A conventional way to derive fluid-dynamical equations of motion from the Boltzmann equation is to expand the momentum distribution function around an isotropic local equilibrium distribution function. However, especially in the early stages of the stages of relativistic heavy-ion collisions, the deviations from equilibrium can be significant, and such an expansion is expected to break down. Nevertheless, one can still derive a fluid-dynamical theory, called anisotropic dissipative fluid dynamics, in terms of an expansion around an anisotropic single-particle distribution function, which incorporates at least parts of the momentum anisotropy. We have constructed the basic formalism for such an expansion in a way that can be systematically improved. We derived the equations of motion of anisotropic dissipative fluid dynamics starting from the Boltzmann equation.
We have then applied the above formalism for the one-dimensional Bjorken expansion, which is in particular relevant for the early stages of a heavy-ion collision. In order to close the equations of motion of anisotropic fluid dynamics, one needs to choose an additional moment of the Boltzmann equation. In this work we considered several choices and compared the corresponding fluid-dynamical solutions to the exact solution of the Boltzmann equation. We were then able to identify the best choice for the moment, and showed that the formalism can be extended to situations where the system is strongly out-of-equilibrium.
We analyzed the directed flow in high-energy heavy-ion collisions in the energy range from 7.7 to 27 GeV as a signature for the softening of the equation state (EOS), i.e. as a signature of the QCD phase transition. These lower-energy collision probe in particular the properties of strongly interacting matter at higher baryon density. We gave a detailed analysis of how directed flow is generated during the collision. In particular, we found that the softening of effective EOS in the overlapping region of the two colliding nuclei, i.e. the reaction stages where the system reaches a high-baryon density state, is needed to explain the observed collapse of the proton directed flow within a hadronic transport approach.
This study was further extended to study the sensitivity of the directed flow on the EOS. The EOS was modified by introducing a new collision term in order to control the pressure of a system by appropriately selecting an azimuthal angle in two-body collisions according to a given EOS. The beam-energy dependence of the directed flow of protons was examined with two different EOS, one with a first-order phase transition and one with a crossover. It was found that our approach yields quite similar results as hydrodynamical predictions on the beam-energy dependence of the directed flow: Transport theory predicts a minimum in the excitation function of the slope of proton directed flow and does indeed yield negative directed flow, if the EOS with a first-order phase transition is employed. Our result strongly suggests that the highest sensitivity for the critical point can be seen at beam energies from 4.7 to 11.5 GeV.
In addition, we have compared numerical solutions of conventional fluid dynamics and with solutions of the Boltzmann equation in several situations resembling the ones encountered in heavy-ion collisions. A most significant technical challenge has been to solve the Boltzmann equation numerically with sufficient numerical accuracy. We believe that we have now reached such accuracy, and that we can make meaningful comparisons with fluid-dynamical solutions. We are currently finishing a first publication of this part of the project, where we analyze the conditions for the applicability of fluid dynamics in describing the space-time evolution of the small amount of matter formed in nuclear collisions.
We have also made several predictions for different flow correlations measured by the ALICE Collaboration at the LHC and STAR Collaboration at RHIC. These predictions are based on earlier works, where we have constructed shear-viscosity parametrizations that are consistent with flow measurements at different collision energies at the LHC and RHIC. We are currently preparing a publication, where we analyze these correlations more thoroughly, and study their impact on constraining the transport properties of strongly interacting matter. Moreover, we have presented several predictions for the new measurements at the LHC, i.e. high-energy lead-lead and xenon-xenon collisions, and showed how these new measurements give more detailed information about the properties of strongly interacting matter.
